The way scrollbars work is pretty inefficient - especially for novice computer users. They offer three (!) methods for scrolling relative to the current position and none of them offers comfortable scrolling speed for all document sizes.
The arrows in the corners are very small compared to the scrollbar's overall size and they are too slow in most cases. Still, a lot of people use the arrows to scroll rather large distances in a document.
The scroll thumb (slider) is too small in big documents and requires too much aiming although it is the fastest and - for moderately-sized documents - a sufficiently accurate way to change your relative position. For very big documents this way of scrolling becomes nearly useless because it moves areas bigger than the visible region.
Scrollbar basic lookA third alternative is to scroll page-wise by clicking on the non-highlighted area. This is often too coarse-grained and sometimes collides with the thumb which may suddenly appear under the mouse cursor, so you have to adjust your mouse position to continue scrolling that way. Inexperienced users sometimes click on the non-highlighted area to quickly get to a specific location.
The following proposal should be more effective, but it requires an initial training phase because it's different from the current way of scrolling.
The arrows at the top and bottom are not buttons, anymore. Also, they are bigger to make it more obvious that they are part of the scrollbar background. The scroll thumb lights up when the mouse is moved anywhere on the scrollbar (not just the knob). The general appearance of the scrollbars is very similar to that of the current mainstream operating systems.
Left-click scrollingPress and hold the left mouse button on any place on the scrollbar and you get into relative scroll mode. To clarify, you can click on the thumb, or on the arrows, or anywhere else on the scrollbar and it will have the same effect. By moving the mouse you can scroll a proportional amount of screen area in the same direction. The mouse movement causes the thumb to be moved at comfortable speed which is independent of the document size. Mouse movement has no effect on the cursor's position (i.e.: it works as if you could move beyond the screen borders).
Right-click scrollingRight-click (tablet users will prefer ALT+left-click) on any place on the scrollbar and you immediately jump to that location in the document. By keeping the mouse button pressed and moving the mouse the thumb will move directly with the mouse pointer as if you had right-clicked there.
When moving the mouse over a scrollbar the cursor becomes a double-arrow (up/down or left/right), so it becomes more obvious that by left-clicking you can move the scroll knob.
Pressing ESC with the mouse button held will bring you back to the last position before scrolling.
This proposal has the disadvantage that the right-click scrolling function is non-obvious and users would have to learn it, so it's not perfect and it could need more work.
I really like your right-mouse-button idea. After reading your description, I played around with different mouse buttons on the scrollbar in my window manager, KDE. Apparently, the MMB works the way your proposed RMB function would work. It's a neat function, and I wish it were the primary one.
In fact, I think that should be the primary function in your setup too, rather than the secondary one. The reason for this is that "absolute scrolling" can be used anywhere, and only becomes difficult in very large documents, in which case its usefulness is limited by the user's own dexterity. For these cases, we need a special function: your proposed "relative scrolling."
As a primary function, relative scrolling is likely to feel limiting (it would be as if the scrollbar had only up and down buttons). We don't want the user to have that impression.
I think we can make the transition to the new scrollbar style even easier for the user if we keep the scroll buttons. That way, anyone who knows how to use scrollbars in other operating systems can use them here. With this setup, nothing is lost if you haven't ascended the learning curve. It just feels like a normal scrollbar. Learning to use the RMB function (relative scrolling) will help the user, but they won't feel like they're missing something without it.
I've said a lot here, and I'm trying to be as clear as possible, but I'm sure it looks better in my head than it will in yours. Push back, ask for clarification, correct me, disagree, whatever. Give me some feedback.